Binary Class AUC. Let \(\mathcal {D}_{\mathcal {Z}}\)  denote the joint data distribution and \(f\)  denote a score function estimating the possibility that an underlying instance belongs to the positive class. In the context of binary class problems, AUC is known to have a clear statistical meaning: it is equivalent to the Wilcoxon Statistics [1]}, namely the possibility that correct ranking takes place between a random paired samples with distinct labels: \({\rm AUC}(f) = \mathbb {P}\big [\Delta (y)\Delta (f) >0 | \mathcal {E}^{(0,1)}\big ] + \frac{1}{2} \mathbb {P}\big [\Delta (f) = 0 | \mathcal {E}^{(0,1)}\big ]= \operatornamewithlimits{\mathbb {E}}_{\mathbf {z}_1 \sim \mathcal {D}_{\mathcal {Z} },\mathbf {z}_2 \sim \mathcal {D}_{\mathcal {Z}}}\big [\mathbf {I}[\Delta (y)\Delta (f) > 0] + \frac{1}{2}\mathbf {I}[\Delta (f) = 0] | \mathcal {E}^{(0,1)}\big ],\) 
where \( \Delta (y) = y_1 - y_2, ~~ \Delta (f) = f(\mathbf {x}_1) - f(\mathbf {x}_2)\) . Note that we adopt the convention [2]}, [3]}, [4]} to score ties with 0.5. See basic notations in the Sec.REF  for \(\mathcal {E}^{(0,1)}\) . Hereafter, our calculations on AUC follows this definition.
